Accelerating Solutions of Nonlinear PDEs Using Machine Learning: A Case Study with the Network Transport Model

BrandonImstepf

University of California, Merced

"Accelerating Solutions of Nonlinear PDEs Using Machine Learning: A Case Study with the Network Transport Model"

Alzheimer’s Disease (AD) is a progressive neurodegenerative disorder affecting approximately 10% of Americans over age 65, leading to memory loss, cognitive decline, and impaired daily function. Disease progression correlates with the spread of tau and amyloid-beta proteins, which aggregate into neurofibrillary tangles. While macroscopic whole-brain network models predict large-scale protein deposition patterns, they lack the specificity to capture individual disease progression. Conversely, microscale neuron-neuron models offer highly detailed biochemical aggregation and transport simulations but are computationally prohibitive for whole-brain parameter inference. In this work, we explore using machine learning to accelerate whole-brain simulations by approximating explicit solutions to the microscopic Two-Neuron Transport Model (TNTM), a partial differential equation describing tau flux along a neuron, incorporating biochemical aggregation, fragmentation, and transport. We simulate a single-edge model across physiological ranges of biochemical parameters and boundary conditions, then compare regression methods with varying levels of interpretability, from neural networks (low) to symbolic regression via PySR (high). Neural networks achieve the lowest error but lack biological insight. Linear and polynomial regression compute rapidly but yield high errors with limited interpretability. Symbolic regression achieves a balance between accuracy and transparency. This work demonstrates the potential of machine learning for computationally scalable AD modeling, opening avenues for patient-specific parameterization using AD data repositories.
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